Option trading forms part of our financial markets. A traded option gives to its owner the right to buy (call option) or to sell (put option) a fixed quantity of assets of a specified stock at a fixed price (exercise or strike price). There are two major types of traded options. One is the American option that can be exercised at any time prior to its expiry date, and the other option, which ca...

Examples of the analytical approximations are provided in Milevsky and Posner 1998 [3] and [4] who compare the relative accuracy of the lognormal or the inverse gamma distribution for approximating the sum of lognormal distributions. Tree based methods were originally proposed by Cox et al 1979 [2] and adopted in Wan 2002 [5]. Monte Carlo methods were first proposed by Boyle 1977 [1] as an alte...

We consider the problem of pricing a contingent claim whose payoff depends on several sources of uncertainty. Using classical assumptions from the Arbitrage Pricing Theory, the theoretical price can be computed as the discounted expected value of future cash flows under the modified risk-neutral information process. Although analytical solutions have been developed in the literature for a few p...

In this text, Fractional Brown Motion theory during random process is applied to research the option pricing problem. Firstly, Fractional Brown Motion theory and actuarial pricing method of option are utilized to derive Black-Scholes formula under Fractional Brown Motion and form corresponding mathematical model to describe option pricing. Secondly, based on BYD stock, estimation model on volat...

We investigate finite element discretizations using functions that are discontinuous in time and continuous in space for European options with local volatility Black-Scholes models. We present an a posteriori error estimate where a user-specified functional of the error is controlled by the inner product of the finite element residual with the solution of a dual problem that involves the densit...

The primary goal of this paper is to price European options in the Merton’s framework with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes a...

Asian options are strongly path-dependent derivatives. Although efficient numerical methods and approximate closedform formulas are available, most lack convergence guarantees. Asian options can also be priced on the lattice. All efficient lattice algorithms keep only a polynomial number of states and use interpolation to compensate for the less than full representation of the states. Let the t...

We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backwar...

A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools. JEL G13